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			<h1 id="firstHeading" class="firstHeading">Nyquist–Shannon sampling theorem</h1>
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<div class="thumbinner" style="width:202px;"><a href="http://en.wikipedia.org/wiki/File:Bandlimited.svg" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/200px-Bandlimited.png" class="thumbimage" height="102" width="200"></a>
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<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Bandlimited.svg" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.1:</b> Hypothetical spectrum of a <b>bandlimited signal</b> as a function of frequency</div>
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<p>The <b>Nyquist–Shannon sampling theorem</b>, after <a href="http://en.wikipedia.org/wiki/Harry_Nyquist" title="Harry Nyquist">Harry Nyquist</a> and <a href="http://en.wikipedia.org/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>, is a fundamental result in the field of <a href="http://en.wikipedia.org/wiki/Information_theory" title="Information theory">information theory</a>, in particular <a href="http://en.wikipedia.org/wiki/Telecommunication" title="Telecommunication">telecommunications</a> and <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a>. <a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling</a> is the process of converting a <a href="http://en.wikipedia.org/wiki/Signal_%28electrical_engineering%29" title="Signal (electrical engineering)" class="mw-redirect">signal</a> (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). <a href="http://en.wikipedia.org/wiki/Claude_Shannon" title="Claude Shannon">Shannon's</a> version of the theorem states:<sup id="cite_ref-Shannon49_0-0" class="reference"><a href="#cite_note-Shannon49-0"><span>[</span>1<span>]</span></a></sup></p>
<blockquote>
<p>If a function <i>x</i>(<i>t</i>) contains no frequencies higher than <i>B</i> <a href="http://en.wikipedia.org/wiki/Hertz" title="Hertz">hertz</a>, it is completely determined by giving its ordinates at a series of points spaced 1/(2<i>B</i>) seconds apart.</p>
</blockquote>
<p>The theorem is commonly called the <b>Nyquist sampling theorem</b>; since it was also discovered independently by <a href="http://en.wikipedia.org/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a>, by <a href="http://en.wikipedia.org/wiki/Vladimir_Kotelnikov" title="Vladimir Kotelnikov">Vladimir Kotelnikov</a>, and by others, it is also known as <b>Nyquist–Shannon–Kotelnikov</b>, <b>Whittaker–Shannon–Kotelnikov</b>, <b>Whittaker–Nyquist–Kotelnikov–Shannon</b>, <b>WKS</b>, etc., sampling theorem, as well as the <b>Cardinal Theorem of Interpolation Theory</b>. It is often referred to simply as <i><b>the sampling theorem</b></i>.</p>
<p>In essence, the theorem shows that a <a href="http://en.wikipedia.org/wiki/Bandlimited" title="Bandlimited" class="mw-redirect">bandlimited</a> <a href="http://en.wikipedia.org/wiki/Analog_signal" title="Analog signal">analog signal</a> that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2<i>B</i> samples per second, where <i>B</i> is the highest <a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">frequency</a> in the original signal. If a signal contains a component at exactly <i>B</i> hertz, then samples spaced at exactly 1/(2<i>B</i>) seconds do not completely determine the signal, Shannon's statement notwithstanding. This <a href="http://en.wikipedia.org/wiki/Necessary_and_sufficient_condition" title="Necessary and sufficient condition">sufficient condition</a> can be weakened, as discussed at <a href="#Sampling_of_non-baseband_signals">Sampling of non-baseband signals</a> below.</p>
<p>More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if <i>x</i>(<i>t</i>) contains no frequencies higher than <i>or equal to</i> <i>B</i>; this condition is equivalent to Shannon's except when the function includes a steady <a href="http://en.wikipedia.org/wiki/Sinusoid" title="Sinusoid" class="mw-redirect">sinusoidal</a> component at exactly frequency <i>B</i>.</p>
<p>The theorem assumes an idealization of any real-world situation, as 
it only applies to signals that are sampled for infinite time; any 
time-limited <i>x</i>(<i>t</i>) cannot be perfectly <a href="http://en.wikipedia.org/wiki/Bandlimited#Bandlimited_versus_timelimited" title="Bandlimited" class="mw-redirect">bandlimited</a>.
 Perfect reconstruction is mathematically possible for the idealized 
model but only an approximation for real-world signals and sampling 
techniques, albeit in practice often a very good one.</p>
<p>The theorem also leads to a formula for reconstruction of the 
original signal. The constructive proof of the theorem leads to an 
understanding of the <a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">aliasing</a> that can occur when a sampling system does not satisfy the conditions of the theorem.</p>
<p>The sampling theorem provides a sufficient condition, but not a necessary one, for perfect reconstruction. The field of <a href="http://en.wikipedia.org/wiki/Compressed_sensing" title="Compressed sensing">compressed sensing</a> provides a stricter sampling condition when the underlying signal is known to be sparse. <a href="http://en.wikipedia.org/wiki/Compressed_sensing" title="Compressed sensing">Compressed sensing</a> specifically yields a sub-Nyquist sampling criterion.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Introduction"><span class="tocnumber">1</span> <span class="toctext">Introduction</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#The_sampling_process"><span class="tocnumber">2</span> <span class="toctext">The sampling process</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Reconstruction"><span class="tocnumber">3</span> <span class="toctext">Reconstruction</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#Practical_considerations"><span class="tocnumber">4</span> <span class="toctext">Practical considerations</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#Aliasing"><span class="tocnumber">5</span> <span class="toctext">Aliasing</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#Application_to_multivariable_signals_and_images"><span class="tocnumber">6</span> <span class="toctext">Application to multivariable signals and images</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="#Downsampling"><span class="tocnumber">7</span> <span class="toctext">Downsampling</span></a></li>
<li class="toclevel-1 tocsection-8"><a href="#Critical_frequency"><span class="tocnumber">8</span> <span class="toctext">Critical frequency</span></a></li>
<li class="toclevel-1 tocsection-9"><a href="#Mathematical_reasoning_for_the_theorem"><span class="tocnumber">9</span> <span class="toctext">Mathematical reasoning for the theorem</span></a></li>
<li class="toclevel-1 tocsection-10"><a href="#Shannon.27s_original_proof"><span class="tocnumber">10</span> <span class="toctext">Shannon's original proof</span></a></li>
<li class="toclevel-1 tocsection-11"><a href="#Sampling_of_non-baseband_signals"><span class="tocnumber">11</span> <span class="toctext">Sampling of non-baseband signals</span></a></li>
<li class="toclevel-1 tocsection-12"><a href="#Nonuniform_sampling"><span class="tocnumber">12</span> <span class="toctext">Nonuniform sampling</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#Beyond_Nyquist"><span class="tocnumber">13</span> <span class="toctext">Beyond Nyquist</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#Historical_background"><span class="tocnumber">14</span> <span class="toctext">Historical background</span></a>
<ul>
<li class="toclevel-2 tocsection-15"><a href="#Other_discoverers"><span class="tocnumber">14.1</span> <span class="toctext">Other discoverers</span></a></li>
<li class="toclevel-2 tocsection-16"><a href="#Why_Nyquist.3F"><span class="tocnumber">14.2</span> <span class="toctext">Why Nyquist?</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-17"><a href="#See_also"><span class="tocnumber">15</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-18"><a href="#Notes"><span class="tocnumber">16</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-19"><a href="#References"><span class="tocnumber">17</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-20"><a href="#External_links"><span class="tocnumber">18</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=1" title="Edit section: Introduction">edit</a>]</span> <span class="mw-headline" id="Introduction">Introduction</span></h2>
<p>A signal or function is bandlimited if it contains no <a href="http://en.wikipedia.org/wiki/Energy_%28signal_processing%29" title="Energy (signal processing)">energy</a> at frequencies higher than some bandlimit or <a href="http://en.wikipedia.org/wiki/Bandwidth_%28signal_processing%29" title="Bandwidth (signal processing)">bandwidth</a> <i>B</i>.
 A signal that is bandlimited is constrained in how rapidly it changes 
in time, and therefore how much detail it can convey in an interval of 
time. The sampling theorem asserts that the uniformly spaced discrete 
samples are a complete representation of the signal if this bandwidth is
 less than half the sampling rate. To formalize these concepts, let <i>x</i>(<i>t</i>) represent a <a href="http://en.wikipedia.org/wiki/Continuous-time" title="Continuous-time" class="mw-redirect">continuous-time</a> signal and <i>X</i>(<i>f</i>) be the continuous <a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of that signal:</p>
<dl>
<dd><img class="tex" alt="X(f)\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) \ e^{- i 2 \pi f t} \ dt." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/070c71166f9752b286a9230217228c0b.png"></dd>
</dl>
<p>The signal <i>x</i>(<i>t</i>) is said to be bandlimited to a one-sided baseband bandwidth, <i>B</i>, if</p>
<dl>
<dd><img class="tex" alt="X(f) = 0 \quad " src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/1483b14b01826a3540179360687b9582.png">&nbsp; for all &nbsp; <span class="texhtml">| <i>f</i> | &gt; <i>B</i>,</span></dd>
</dl>
<p>or, equivalently, <span style="white-space:nowrap;">supp(<i>X</i>) ⊆ [−<i>B</i>, <i>B</i>]</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup> Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate <i>f<sub>s</sub></i> (in samples per unit time) is:</p>
<dl>
<dd><img class="tex" alt="f_s &gt; 2 B.\!" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/b5b0f39313f7a1eac6c9e63c8edcf7c3.png"></dd>
</dl>
<p>The quantity 2<i>B</i> is called the <i><a href="http://en.wikipedia.org/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a></i> and is a property of the bandlimited signal, while <i>f<sub>s</sub></i>/2 is called the <i><a href="http://en.wikipedia.org/wiki/Nyquist_frequency" title="Nyquist frequency">Nyquist frequency</a></i> and is a property of this sampling system.</p>
<p>The time interval between successive samples is referred to as the <i>sampling interval</i>:</p>
<dl>
<dd><img class="tex" alt="T\ \stackrel{\mathrm{def}}{=}\ \frac{1}{f_s},\," src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/09a21f1ddc505a3c477d014d280cca87.png"></dd>
</dl>
<p>and the samples of <i>x</i>(<i>t</i>) are denoted by:</p>
<dl>
<dd><img class="tex" alt="x[n] = x(nT),\!" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/7758b0866f5f27b4840841bab99bbfed.png"></dd>
</dl>
<p>where <i>n</i> is an integer. The sampling theorem leads to a procedure for reconstructing the original <i>x</i>(<i>t</i>) from the samples and states sufficient conditions for such a reconstruction to be exact.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=2" title="Edit section: The sampling process">edit</a>]</span> <span class="mw-headline" id="The_sampling_process">The sampling process</span></h2>
<p>The theorem describes two processes in <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a>: a <a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">sampling</a> process, in which a <a href="http://en.wikipedia.org/wiki/Continuous_time" title="Continuous time" class="mw-redirect">continuous time</a> signal is converted to a <a href="http://en.wikipedia.org/wiki/Discrete_time" title="Discrete time">discrete time</a> signal, and a reconstruction process, in which the original continuous signal is recovered from the discrete time signal.</p>
<p>The continuous signal varies over <i>time</i> (or <i>space</i> in a digitized <a href="http://en.wikipedia.org/wiki/Image" title="Image">image</a>,
 or another independent variable in some other application) and the 
sampling process is performed by measuring the continuous signal's value
 every <i>T</i> units of time (or space), which is called the <i>sampling interval</i>. Sampling results in a sequence of numbers, called <i>samples</i>,
 to represent the original signal. Each sample value is associated with 
the instant in time when it was measured. The reciprocal of the sampling
 interval (1/<i>T</i>) is the <a href="http://en.wikipedia.org/wiki/Sampling_frequency" title="Sampling frequency" class="mw-redirect">sampling frequency</a> denoted <i>f<sub>s</sub></i>, which is measured in samples per unit of time. If <i>T</i> is expressed in <a href="http://en.wikipedia.org/wiki/Second" title="Second">seconds</a>, then <i>f<sub>s</sub></i> is expressed in <a href="http://en.wikipedia.org/wiki/Hertz" title="Hertz">hertz</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=3" title="Edit section: Reconstruction">edit</a>]</span> <span class="mw-headline" id="Reconstruction">Reconstruction</span></h2>
<p>Reconstruction of the original signal is an <a href="http://en.wikipedia.org/wiki/Interpolation" title="Interpolation">interpolation</a> process that mathematically defines a continuous-time signal <i>x</i>(<i>t</i>) from the discrete samples <i>x</i>[<i>n</i>] and at times in between the sample instants <i>nT</i>.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Sinc_function_%28normalized%29.svg" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/300px-Sinc_function_normalized.png" class="thumbimage" height="206" width="300"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Sinc_function_%28normalized%29.svg" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.2:</b> The normalized <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>: sin(π<i>x</i>) / (π<i>x</i>) ... showing the central peak at <i>x</i>= 0, and zero-crossings at the other integer values of <i>x</i>.</div>
</div>
</div>
<ul>
<li><b>The procedure:</b> Each sample value is multiplied by the <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>
 scaled so that the zero-crossings of the sinc function occur at the 
sampling instants and that the sinc function's central point is shifted 
to the time of that sample, <i>nT</i>. All of these shifted and scaled 
functions are then added together to recover the original signal. The 
scaled and time-shifted sinc functions are <a href="http://en.wikipedia.org/wiki/Continuous_function" title="Continuous function">continuous</a>
 making the sum of these also continuous, so the result of this 
operation is a continuous signal. This procedure is represented by the <a href="http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a>.</li>
</ul>
<ul>
<li><b>The condition:</b> The signal obtained from this reconstruction 
process can have no frequencies higher than one-half the sampling 
frequency. According to the theorem, the reconstructed signal will match
 the original signal provided that the original signal contains no 
frequencies at or above this limit. This condition is called the <i>Nyquist criterion</i>, or sometimes the <i>Raabe condition.</i></li>
</ul>
<p>If the original signal contains a frequency component equal to 
one-half the sampling rate, the condition is not satisfied. The 
resulting reconstructed signal may have a component at that frequency, 
but the amplitude and phase of that component generally will not match 
the original component.</p>
<p>This reconstruction or interpolation using sinc functions is not the 
only interpolation scheme. Indeed, it is impossible in practice because 
it requires summing an infinite number of terms. However, it is the 
interpolation method that in theory exactly reconstructs <i>any</i> given bandlimited <i>x</i>(<i>t</i>) with <i>any</i> bandlimit <i>B</i> &lt; 1/(2<i>T</i>); any other method that does so is formally equivalent to it.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=4" title="Edit section: Practical considerations">edit</a>]</span> <span class="mw-headline" id="Practical_considerations">Practical considerations</span></h2>
<p>A few consequences can be drawn from the theorem:</p>
<ul>
<li>If the highest frequency <i>B</i> in the original signal is known, 
the theorem gives the lower bound on the sampling frequency for which 
perfect reconstruction can be assured. This lower bound to the sampling 
frequency, 2<i>B</i>, is called the <a href="http://en.wikipedia.org/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a>.</li>
</ul>
<ul>
<li>If instead the sampling frequency is known, the theorem gives us an upper bound for frequency components, <i>B</i>&lt;<i>f<sub>s</sub></i>/2, of the signal to allow for perfect reconstruction. This upper bound is the <a href="http://en.wikipedia.org/wiki/Nyquist_frequency" title="Nyquist frequency">Nyquist frequency</a>, denoted <i>f<sub>N</sub></i>.</li>
</ul>
<ul>
<li>Both of these cases imply that the signal to be sampled must be <a href="http://en.wikipedia.org/wiki/Bandlimited" title="Bandlimited" class="mw-redirect">bandlimited</a>;
 that is, any component of this signal which has a frequency above a 
certain bound should be zero, or at least sufficiently close to zero to 
allow us to neglect its influence on the resulting reconstruction. In 
the first case, the condition of bandlimitation of the sampled signal 
can be accomplished by assuming a model of the signal which can be 
analysed in terms of the frequency components it contains; for example, 
sounds that are made by a speaking human normally contain very small 
frequency components at or above 10&nbsp;kHz and it is then sufficient 
to sample such an audio signal with a sampling frequency of at least 
20&nbsp;kHz. For the second case, we have to assure that the sampled 
signal is bandlimited such that frequency components at or above half of
 the sampling frequency can be neglected. This is usually accomplished 
by means of a suitable low-pass filter; for example, if it is desired to
 sample speech waveforms at 8&nbsp;kHz, the signals should first be 
lowpass filtered to below 4&nbsp;kHz.</li>
</ul>
<ul>
<li>In practice, neither of the two statements of the sampling theorem 
described above can be completely satisfied, and neither can the 
reconstruction formula be precisely implemented. The reconstruction 
process that involves scaled and delayed <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc functions</a> can be described as <i>ideal</i>.
 It cannot be realized in practice since it implies that each sample 
contributes to the reconstructed signal at almost all time points, 
requiring summing an infinite number of terms. Instead, some type of 
approximation of the sinc functions, finite in length, has to be used. 
The error that corresponds to the sinc-function approximation is 
referred to as <i>interpolation error</i>. Practical <a href="http://en.wikipedia.org/wiki/Digital-to-analog_converter" title="Digital-to-analog converter">digital-to-analog converters</a> produce neither scaled and delayed sinc functions nor ideal <a href="http://en.wikipedia.org/wiki/Dirac_delta" title="Dirac delta" class="mw-redirect">impulses</a> (that if ideally low-pass filtered would yield the original signal), but a sequence of scaled and delayed <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular pulses</a>. This practical <a href="http://en.wikipedia.org/wiki/Piecewise-constant" title="Piecewise-constant" class="mw-redirect">piecewise-constant</a> output can be modeled as a <a href="http://en.wikipedia.org/wiki/Zero-order_hold" title="Zero-order hold">zero-order hold</a>
 filter driven by the sequence of scaled and delayed dirac impulses 
referred to in the mathematical basis section below. A shaping filter is
 sometimes used after the DAC with zero-order hold to make a better 
overall approximation.</li>
</ul>
<ul>
<li>Furthermore, in practice, a signal can never be perfectly 
bandlimited, since ideal "brick-wall" filters cannot be realized. All 
practical filters can only attenuate frequencies outside a certain 
range, not remove them entirely. In addition to this, a "time-limited" 
signal can never be bandlimited. This means that even if an ideal 
reconstruction could be made, the reconstructed signal would not be 
exactly the original signal. The error that corresponds to the failure 
of bandlimitation is referred to as <i>aliasing</i>.</li>
</ul>
<ul>
<li>The sampling theorem does not say what happens when the conditions 
and procedures are not exactly met, but its proof suggests an analytical
 framework in which the non-ideality can be studied. A designer of a 
system that deals with sampling and reconstruction processes needs a 
thorough understanding of the signal to be sampled, in particular its 
frequency content, the sampling frequency, how the signal is 
reconstructed in terms of interpolation, and the requirement for the 
total reconstruction error, including aliasing, sampling, interpolation 
and other errors. These properties and parameters may need to be 
carefully tuned in order to obtain a useful system.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=5" title="Edit section: Aliasing">edit</a>]</span> <span class="mw-headline" id="Aliasing">Aliasing</span></h2>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">Aliasing</a></div>
<p>The <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a> shows that the samples, <i>x</i>[<i>n</i>]=<i>x</i>(<i>nT</i>), of function <i>x</i>(<i>t</i>) are sufficient to create a <a href="http://en.wikipedia.org/wiki/Periodic_summation" title="Periodic summation">periodic summation</a> of function <i>X</i>(<i>f</i>). The result is:</p>
<dl>
<dd>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none;">
<tbody><tr>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0;"><img class="tex" alt="X_s(f)\ \stackrel{\mathrm{def}}{=} \sum_{k=-\infty}^{\infty} X\left(f - k f_s\right) = T \sum_{n=-\infty}^{\infty} x(nT)\ e^{-i 2\pi n T f}." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/9215b5f448856ba28cd90c60944a7a71.png"></p>
</td>
<td style="vertical-align: middle; width: 99%; border: medium none; padding: 0.08em;">
<p style="margin:0;"></p>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none; width: 99%;">
<tbody><tr>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
<td style="width: 100%; border: medium none; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
</tr>
<tr>
<td style="border-left: medium none; border-width: 0px medium medium; border-style: none; border-color: rgb(229, 229, 229) -moz-use-text-color -moz-use-text-color; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
</tr>
</tbody></table>
</td>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0pt;"><b>(<cite id="math_Eq.1"></cite><span class="reference plainlinksneverexpand"><cite id="math_Eq.1"><a href="#equation_Eq.1">Eq.1</a></cite><b><cite id="math_Eq.1"></cite>)</b></span></b></p>
</td>
</tr>
</tbody></table>
</dd>
</dl>
<p>As depicted in Figures 3, 4, and 8, copies of <i>X</i>(<i>f</i>) are shifted by multiples of <i>f<sub>s</sub></i> and combined by addition.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:NonoverlappedSpectrum.png" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/400px-NonoverlappedSpectrum.png" class="thumbimage" height="100" width="400"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:NonoverlappedSpectrum.png" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.3:</b> Hypothetical spectrum of a properly sampled bandlimited signal (blue) and images (green) that do not overlap. A "<i>brick-wall</i>" low-pass filter can remove the images and leave the original spectrum, thus recovering the original signal from the samples.</div>
</div>
</div>
<p>If the sampling condition is not satisfied, adjacent copies overlap, 
and it is not possible in general to discern an unambiguous <i>X</i>(<i>f</i>). Any frequency component above <i>f<sub>s</sub></i>/2 is indistinguishable from a lower-frequency component, called an <i>alias</i>,
 associated with one of the copies. The reconstruction technique 
described below produces the alias, rather than the original component, 
in such cases.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:AliasedSpectrum.png" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/400px-AliasedSpectrum.png" class="thumbimage" height="271" width="400"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:AliasedSpectrum.png" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.4 Top:</b> Hypothetical spectrum of an insufficiently sampled bandlimited signal (blue), <i>X</i>(<i>f</i>), where the images (green) overlap. These overlapping edges or "<i>tails</i>" of the images add, creating a spectrum unlike the original. <b>Bottom:</b> Hypothetical spectrum of a marginally sufficiently sampled bandlimited signal (blue), <i>X</i><sub>A</sub>(<i>f</i>), where the images (green) narrowly do not overlap. But the overall sampled spectrum of <i>X</i><sub>A</sub>(<i>f</i>) is identical to the overall inadequately sampled spectrum of <i>X</i>(<i>f</i>) (top) because the sum of baseband and images are the same in both cases. The discrete sampled signals <i>x</i><sub>A</sub>[<i>n</i>] and <i>x</i>[<i>n</i>]
 are also identical. It is not possible, just from examining the spectra
 (or the sampled signals), to tell the two situations apart. If this 
were an audio signal, <i>x</i><sub>A</sub>[<i>n</i>] and <i>x</i>[<i>n</i>] would sound the same and the presumed "<i>properly</i>" sampled <i>x</i><sub>A</sub>[<i>n</i>] would be the <i>alias</i> of <i>x</i>[<i>n</i>] since the spectrum <i>X</i><sub>A</sub>(<i>f</i>) masquerades as the spectrum <i>X</i>(<i>f</i>).</div>
</div>
</div>
<p>For a sinusoidal component of exactly half the sampling frequency, 
the component will in general alias to another sinusoid of the same 
frequency, but with a different phase and amplitude.</p>
<p>To prevent or reduce aliasing, two things can be done:</p>
<ol>
<li>Increase the sampling rate, to above twice some or all of the frequencies that are aliasing.</li>
<li>Introduce an <a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a> or make the anti-aliasing filter more stringent.</li>
</ol>
<p>The anti-aliasing filter is to restrict the bandwidth of the signal 
to satisfy the condition for proper sampling. Such a restriction works 
in theory, but is not precisely satisfiable in reality, because 
realizable filters will always allow some <i>leakage</i> of high frequencies. However, the leakage energy can be made small enough so that the aliasing effects are negligible.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=6" title="Edit section: Application to multivariable signals and images">edit</a>]</span> <span class="mw-headline" id="Application_to_multivariable_signals_and_images">Application to multivariable signals and images</span></h2>
<div class="thumb tleft">
<div class="thumbinner" style="width:152px;"><a href="http://en.wikipedia.org/wiki/File:Moire_pattern_of_bricks_small.jpg" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/150px-Moire_pattern_of_bricks_small.jpg" class="thumbimage" height="183" width="150"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Moire_pattern_of_bricks_small.jpg" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.5:</b> Subsampled image showing a <a href="http://en.wikipedia.org/wiki/Moir%C3%A9_pattern" title="Moiré pattern">Moiré pattern</a></div>
</div>
</div>
<div class="thumb tleft">
<div class="thumbinner" style="width:152px;"><a href="http://en.wikipedia.org/wiki/File:Moire_pattern_of_bricks.jpg" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/150px-Moire_pattern_of_bricks.jpg" class="thumbimage" height="182" width="150"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Moire_pattern_of_bricks.jpg" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.6:</b> See full size image</div>
</div>
</div>
<p>The sampling theorem is usually formulated for functions of a single 
variable. Consequently, the theorem is directly applicable to 
time-dependent signals and is normally formulated in that context. 
However, the sampling theorem can be extended in a straightforward way 
to functions of arbitrarily many variables. Grayscale images, for 
example, are often represented as two-dimensional arrays (or matrices) 
of real numbers representing the relative intensities of <a href="http://en.wikipedia.org/wiki/Pixel" title="Pixel">pixels</a>
 (picture elements) located at the intersections of row and column 
sample locations. As a result, images require two independent variables,
 or indices, to specify each pixel uniquely — one for the row, and one 
for the column.</p>
<p>Color images typically consist of a composite of three separate 
grayscale images, one to represent each of the three primary colors — 
red, green, and blue, or <i>RGB</i> for short. Other colorspaces using 
3-vectors for colors include HSV, LAB, XYZ, etc. Some colorspaces such 
as cyan, magenta, yellow, and black (CMYK) may represent color by four 
dimensions. All of these are treated as <a href="http://en.wikipedia.org/wiki/Vector-valued_function" title="Vector-valued function">vector-valued functions</a> over a two-dimensional sampled domain.</p>
<p>Similar to one-dimensional discrete-time signals, images can also 
suffer from aliasing if the sampling resolution, or pixel density, is 
inadequate. For example, a digital photograph of a striped shirt with 
high frequencies (in other words, the distance between the stripes is 
small), can cause aliasing of the shirt when it is sampled by the 
camera's <a href="http://en.wikipedia.org/wiki/Image_sensor" title="Image sensor">image sensor</a>. The aliasing appears as a <a href="http://en.wikipedia.org/wiki/Moir%C3%A9_pattern" title="Moiré pattern">moiré pattern</a>.
 The "solution" to higher sampling in the spatial domain for this case 
would be to move closer to the shirt, use a higher resolution sensor, or
 to optically blur the image before acquiring it with the sensor.</p>
<p>Another example is shown to the left in the brick patterns. The top 
image shows the effects when the sampling theorem's condition is not 
satisfied. When software rescales an image (the same process that 
creates the thumbnail shown in the lower image) it, in effect, runs the 
image through a low-pass filter first and then <a href="http://en.wikipedia.org/wiki/Downsampling" title="Downsampling">downsamples</a> the image to result in a smaller image that does not exhibit the <a href="http://en.wikipedia.org/wiki/Moir%C3%A9_pattern" title="Moiré pattern">moiré pattern</a>. The top image is what happens when the image is downsampled without low-pass filtering: aliasing results.</p>
<p>The application of the sampling theorem to images should be made with
 care. For example, the sampling process in any standard image sensor 
(CCD or CMOS camera) is relatively far from the ideal sampling which 
would measure the image intensity at a single point. Instead these 
devices have a relatively large sensor area at each sample point in 
order to obtain sufficient amount of light. In other words, any detector
 has a finite-width <a href="http://en.wikipedia.org/wiki/Point_spread_function" title="Point spread function">point spread function</a>.
 The analog optical image intensity function which is sampled by the 
sensor device is not in general bandlimited, and the non-ideal sampling 
is itself a useful type of low-pass filter, though not always sufficient
 to remove enough high frequencies to sufficiently reduce aliasing. When
 the area of the sampling spot (the size of the pixel sensor) is not 
large enough to provide sufficient <a href="http://en.wikipedia.org/wiki/Anti-aliasing" title="Anti-aliasing" class="mw-redirect">anti-aliasing</a>, a separate <a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a>
 (optical low-pass filter) is typically included in a camera system to 
further blur the optical image. Despite images having these problems in 
relation to the sampling theorem, the theorem can be used to describe 
the basics of down and up sampling of images.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=7" title="Edit section: Downsampling">edit</a>]</span> <span class="mw-headline" id="Downsampling">Downsampling</span></h2>
<p>When a signal is <a href="http://en.wikipedia.org/wiki/Downsampling" title="Downsampling">downsampled</a>,
 the sampling theorem can be invoked via the artifice of resampling a 
hypothetical continuous-time reconstruction. The Nyquist criterion must 
still be satisfied with respect to the new lower sampling frequency in 
order to avoid aliasing. To meet the requirements of the theorem, the 
signal must usually pass through a <a href="http://en.wikipedia.org/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>
 of appropriate cutoff frequency as part of the downsampling operation. 
This low-pass filter, which prevents aliasing, is called an <a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=8" title="Edit section: Critical frequency">edit</a>]</span> <span class="mw-headline" id="Critical_frequency">Critical frequency</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:CriticalFrequencyAliasing.svg" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/220px-CriticalFrequencyAliasing.png" class="thumbimage" height="176" width="220"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:CriticalFrequencyAliasing.svg" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.7:</b> A family of sinusoids at the critical frequency, all 
having the same sample sequences of alternating +1 and –1. That is, they
 all are aliases of each other, even though their frequency is not above
 half the sample rate.</div>
</div>
</div>
<p>To illustrate the necessity of <i>f<sub>s</sub></i> &gt; 2<i>B</i>, consider the sinusoid:</p>
<dl>
<dd><img class="tex" alt="x(t) = \cos(2 \pi B t + \theta )\ = \ \cos(2 \pi B t)\cos(\theta ) - \sin(2 \pi B t)\sin(\theta )." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/40418bb3e8a8918f0991a8483a46a31b.png"></dd>
</dl>
<p>With <i>f<sub>s</sub></i> = 2<i>B</i> or equivalently <i>T</i> = 1/(2<i>B</i>), the samples are given by:</p>
<dl>
<dd><img class="tex" alt="
\begin{align}
x(nT) &amp;= \cos(\pi n)\cos(\theta ) - \underbrace{\sin(\pi n)}_{0}\sin(\theta ) = \cos(\pi n)\cos(\theta ).
\end{align}
" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/dfbe85437973843efbf33e5d24fa0e93.png"></dd>
</dl>
<p>Those samples cannot be distinguished from the samples of:</p>
<dl>
<dd><img class="tex" alt="x_A(t) = \cos(2 \pi B t)\cos(\theta ).\," src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/5f9b9857f48548e657658a4d6e6afe91.png"></dd>
</dl>
<p>But for any θ such that sin(θ) ≠ 0, <i>x</i>(<i>t</i>) and <i>x<sub>A</sub></i>(<i>t</i>) have different amplitudes and different phase. This and other ambiguities are the reason for the <i>strict</i> inequality of the sampling theorem's condition.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=9" title="Edit section: Mathematical reasoning for the theorem">edit</a>]</span> <span class="mw-headline" id="Mathematical_reasoning_for_the_theorem">Mathematical reasoning for the theorem</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:ReconstructFilter.png" class="image"><img alt="" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/300px-ReconstructFilter.png" class="thumbimage" height="176" width="300"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:ReconstructFilter.png" class="internal" title="Enlarge"><img src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
<b>Fig.8:</b> Spectrum, <i>X<sub>s</sub></i>(<i>f</i>), of a properly sampled bandlimited signal (blue) and images (green) that do not overlap. A <i>brick-wall</i> low-pass filter, <i>H</i>(<i>f</i>), removes the images, leaves the original spectrum, <i>X</i>(<i>f</i>), and recovers the original signal from the samples.</div>
</div>
</div>
<p>From Figures 3 and 8, it is apparent that when there is no overlap of the copies (aka "images") of <i>X</i>(<i>f</i>), the <i>k</i>&nbsp;=&nbsp;0 term of <i>X<sub>s</sub></i>(<i>f</i>) can be recovered by the product:</p>
<dl>
<dd><img class="tex" alt="X(f) = H(f) \cdot X_s(f),\," src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/f39c77e3fdbda5e854ad0545ef36116f.png">&nbsp; &nbsp; &nbsp; where:</dd>
</dl>
<dl>
<dd><img class="tex" alt="H(f) = \begin{cases}1 &amp; |f| &lt; B \\ 0 &amp; |f| &gt; f_s - B. \end{cases}" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/1d3b5ea45c11ce830b4cc185b786a492.png"></dd>
</dl>
<p><i>H</i>(<i>f</i>) need not be precisely defined in the region <span style="white-space:nowrap;">[<i>B</i>, <i>f<sub>s</sub></i> − <i>B</i>]</span> because <i>X</i><sub>s</sub>(<i>f</i>) is zero in that region. However, the worst case is when <i>B</i>&nbsp;=&nbsp;<i>f<sub>s</sub></i>/2, the Nyquist frequency. A function that is sufficient for that and all less severe cases is:</p>
<dl>
<dd><img class="tex" alt="H(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) = \begin{cases}1 &amp; |f| &lt; \frac{f_s}{2} \\ 0 &amp; |f| &gt; \frac{f_s}{2}, \end{cases}" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/2c9f86c759ce3f65422fe5ebf9c3507f.png"></dd>
</dl>
<p>where rect(<i>u</i>) is the <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular function</a>.</p>
<p>Therefore:</p>
<dl>
<dd><img class="tex" alt="X(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) \cdot X_s(f)\ " src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/33feaf80cecbb731faeefb29d33e8418.png">
<dl>
<dd>
<dl>
<dd><img class="tex" alt=" = \mathrm{rect} (Tf) \cdot T \sum_{n=-\infty}^{\infty} x(nT)\ e^{-i 2\pi n T f}" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/9a47dc2c3f360258b2cb5dc3d80018f7.png">&nbsp; &nbsp; &nbsp; (from &nbsp;<cite id="equation_Eq.1" style="font-style: normal;"><b><a href="#math_Eq.1">Eq.1</a></b></cite>, above).</dd>
<dd><img class="tex" alt=" = T \sum_{n=-\infty}^{\infty} x(nT)\cdot \mathrm{rect} (Tf) \cdot e^{-i 2\pi n T f}." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/a089283876fe263cd2472cb74e5efb9d.png"></dd>
</dl>
</dd>
</dl>
</dd>
</dl>
<p>The original function that was sampled can be recovered by an inverse Fourier transform:</p>
<dl>
<dd><img class="tex" alt="x(t) = \mathcal{F}^{-1}\left \{ T \sum_{n=-\infty}^{\infty} x(nT)\cdot \mathrm{rect} (Tf) \cdot e^{-i 2\pi n T f}\right\}" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/a0c8a1f6585e6e38fc9d3e132ecf3076.png">
<dl>
<dd>
<dl>
<dd><img class="tex" alt="= T \sum_{n=-\infty}^{\infty} x(nT)\cdot \underbrace{\mathcal{F}^{-1}\left \{ \mathrm{rect}(Tf) \cdot e^{-i 2\pi n T f}\right\}}_{\frac{1}{T}\cdot \mathrm{sinc} \left( \frac{t - nT}{T}\right)}" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/c74599edbcb071f2cf7286b85e76b530.png">&nbsp; &nbsp; &nbsp;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>3<span>]</span></a></sup></dd>
<dd><img class="tex" alt="= \sum_{n=-\infty}^{\infty} x(nT)\cdot \mathrm{sinc} \left( \frac{t - nT}{T}\right)," src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/a1cbaab28943708fda1fb2f3ae799144.png"></dd>
</dl>
</dd>
</dl>
</dd>
</dl>
<p>which is the <a href="http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a>. It shows explicitly how the samples, <i>x</i>(<i>nT</i>), can be combined to reconstruct <i>x</i>(<i>t</i>).</p>
<ul>
<li>From Figure 8, it is clear that larger-than-necessary values of <i>f<sub>s</sub></i> (smaller values of <i>T</i>), called <i>oversampling</i>, have no effect on the outcome of the reconstruction and have the benefit of leaving room for a <i>transition band</i> in which <i>H</i>(<i>f</i>) is free to take intermediate values. <a href="http://en.wikipedia.org/wiki/Undersampling" title="Undersampling">Undersampling</a>, which causes aliasing, is not in general a reversible operation.</li>
<li>Theoretically, the interpolation formula can be implemented as a <a href="http://en.wikipedia.org/wiki/Low_pass_filter" title="Low pass filter" class="mw-redirect">low pass filter</a>, whose impulse response is sinc(<i>t</i>/<i>T</i>) and whose input is <img class="tex" alt="\textstyle\sum_{n=-\infty}^{\infty} x(nT)\cdot \delta(t - nT)," src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/f87182a0f7d9cccb2c4d76bfb0e8b0a6.png"> which is a <a href="http://en.wikipedia.org/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a> function modulated by the signal samples. Practical <a href="http://en.wikipedia.org/wiki/Digital-to-analog_converter" title="Digital-to-analog converter">digital-to-analog converters</a> (DAC) implement an approximation like the <a href="http://en.wikipedia.org/wiki/Zero-order_hold" title="Zero-order hold">zero-order hold</a>. In that case, oversampling can reduce the approximation error.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=10" title="Edit section: Shannon's original proof">edit</a>]</span> <span class="mw-headline" id="Shannon.27s_original_proof">Shannon's original proof</span></h2>
<p>The original proof presented by Shannon is elegant and quite brief, 
but it offers less intuitive insight into the subtleties of aliasing, 
both unintentional and intentional. Quoting Shannon's original paper, 
which uses <i>f</i> for the function, <i>F</i> for the spectrum, and <i>W</i> for the bandwidth limit:</p>
<dl>
<dd>Let <span class="texhtml"><i>F</i>(ω)</span> be the spectrum of <span class="texhtml"><i>f</i>(<i>t</i>)</span>. Then</dd>
</dl>
<dl>
<dd>
<dl>
<dd>
<table>
<tbody><tr>
<td><img class="tex" alt="f(t)\," src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/19cc0601bb5dc8fdd4098eb03f50e0b9.png"></td>
<td><img class="tex" alt="= {1 \over 2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t}\;d\omega \ " src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/054eb28f86d6138f3a1cf98f033b57ae.png"></td>
</tr>
<tr>
<td></td>
<td><img class="tex" alt="= {1 \over 2\pi} \int_{-2\pi W}^{2\pi W} F(\omega) e^{i\omega t}\;d\omega \ " src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/367be137138f38b331c589b38d378e94.png"></td>
</tr>
</tbody></table>
</dd>
</dl>
</dd>
</dl>
<dl>
<dd>since <span class="texhtml"><i>F</i>(ω)</span> is assumed to be zero outside the band <i>W</i>. If we let</dd>
</dl>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="t = {n \over {2W}}" src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/93a1573f491b1c5ae0959cc49c9d0ce8.png"></dd>
</dl>
</dd>
</dl>
<dl>
<dd>where <i>n</i> is any positive or negative integer, we obtain</dd>
</dl>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="f \left({n \over {2W}} \right) = {1 \over 2\pi} \int_{-2\pi W}^{2\pi W} F(\omega) e^{i\omega {n \over {2W}}}\;d\omega." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/9763e01d6c7fee65666f9bb7786f2997.png"></dd>
</dl>
</dd>
</dl>
<dl>
<dd>On the left are values of <span class="texhtml"><i>f</i>(<i>t</i>)</span> at the sampling points. The integral on the right will be recognized as essentially the <i>n</i>th coefficient in a Fourier-series expansion of the function <span class="texhtml"><i>F</i>(ω)</span>, taking the interval –<i>W</i> to <i>W</i> as a fundamental period. This means that the values of the samples <span class="texhtml"><i>f</i>(<i>n</i> / 2<i>W</i>)</span> determine the Fourier coefficients in the series expansion of <span class="texhtml"><i>F</i>(ω)</span>. Thus they determine <span class="texhtml"><i>F</i>(ω)</span>, since <span class="texhtml"><i>F</i>(ω)</span> is zero for frequencies greater than <i>W</i>, and for lower frequencies <span class="texhtml"><i>F</i>(ω)</span> is determined if its Fourier coefficients are determined. But <span class="texhtml"><i>F</i>(ω)</span> determines the original function <span class="texhtml"><i>f</i>(<i>t</i>)</span> completely, since a function is determined if its spectrum is known. Therefore the original samples determine the function <span class="texhtml"><i>f</i>(<i>t</i>)</span> completely.</dd>
</dl>
<p>Shannon's proof of the theorem is complete at that point, but he goes
 on to discuss reconstruction via sinc functions, what we now call the <a href="http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a> as discussed above. He does not derive or prove the properties of the <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>, but these would have been familiar to engineers reading his works at the time, since the Fourier pair relationship between <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rect</a> (the rectangular function) and sinc was well known. Quoting Shannon:</p>
<dl>
<dd>Let <span class="texhtml"><i>x</i><sub><i>n</i></sub></span> be the <i>n</i>th sample. Then the function <span class="texhtml"><i>f</i>(<i>t</i>)</span> is represented by:</dd>
</dl>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="f(t) = \sum_{n=-\infty}^{\infty}x_n{\sin \pi(2Wt-n) \over \pi(2Wt-n)}." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/e13f4f2bbc651932062b860aa756a7c4.png"></dd>
</dl>
</dd>
</dl>
<p>As in the other proof, the existence of the Fourier transform of the 
original signal is assumed, so the proof does not say whether the 
sampling theorem extends to bandlimited stationary random processes.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=11" title="Edit section: Sampling of non-baseband signals">edit</a>]</span> <span class="mw-headline" id="Sampling_of_non-baseband_signals">Sampling of non-baseband signals</span></h2>
<p>As discussed by Shannon:<sup id="cite_ref-Shannon49_0-1" class="reference"><a href="#cite_note-Shannon49-0"><span>[</span>1<span>]</span></a></sup></p>
<dl>
<dd>
<blockquote>
<p>A similar result is true if the band does not start at zero frequency
 but at some higher value, and can be proved by a linear translation 
(corresponding physically to <a href="http://en.wikipedia.org/wiki/Single-sideband_modulation" title="Single-sideband modulation">single-sideband modulation</a>) of the zero-frequency case. In this case the elementary pulse is obtained from sin(<i>x</i>)/<i>x</i> by single-side-band modulation.</p>
</blockquote>
</dd>
</dl>
<p>That is, a sufficient no-loss condition for sampling <a href="http://en.wikipedia.org/wiki/Signal_%28information_theory%29" title="Signal (information theory)" class="mw-redirect">signals</a> that do not have <a href="http://en.wikipedia.org/wiki/Baseband" title="Baseband">baseband</a> components exists that involves the <i>width</i> of the non-zero frequency interval as opposed to its highest frequency component. See <i><a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling (signal processing)</a></i> for more details and examples.</p>
<p>A bandpass condition is that <i>X</i>(<i>f</i>) = 0, for all nonnegative <i>f</i> outside the open band of frequencies:</p>
<dl>
<dd>
<dl>
<dd><img class="tex" alt=" \left(\frac{N}2f_\mathrm{s},\frac{N+1}2f_\mathrm{s}\right), " src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/c9f147585a2f9e537cf4976ac9a3ad11.png"></dd>
</dl>
</dd>
</dl>
<p>for some nonnegative integer <i>N</i>. This formulation includes the normal baseband condition as the case <i>N</i>=0.</p>
<p>The corresponding interpolation function is the impulse response of an ideal brick-wall <a href="http://en.wikipedia.org/wiki/Bandpass_filter" title="Bandpass filter" class="mw-redirect">bandpass filter</a> (as opposed to the ideal <a href="http://en.wikipedia.org/wiki/Brick-wall_filter" title="Brick-wall filter" class="mw-redirect">brick-wall</a> <a href="http://en.wikipedia.org/wiki/Lowpass_filter" title="Lowpass filter" class="mw-redirect">lowpass filter</a>
 used above) with cutoffs at the upper and lower edges of the specified 
band, which is the difference between a pair of lowpass impulse 
responses:</p>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="(N+1)\,\operatorname{sinc} \left(\frac{(N+1)t}T\right) - N\,\operatorname{sinc}\left( \frac{Nt}T \right)." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/703bf34fa70a1d62ebeb7c8c487be94a.png"></dd>
</dl>
</dd>
</dl>
<p>Other generalizations, for example to signals occupying multiple 
non-contiguous bands, are possible as well. Even the most generalized 
form of the sampling theorem does not have a provably true converse. 
That is, one cannot conclude that information is necessarily lost just 
because the conditions of the sampling theorem are not satisfied; from 
an engineering perspective, however, it is generally safe to assume that
 if the sampling theorem is not satisfied then information will most 
likely be lost.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=12" title="Edit section: Nonuniform sampling">edit</a>]</span> <span class="mw-headline" id="Nonuniform_sampling">Nonuniform sampling</span></h2>
<p>The sampling theory of Shannon can be generalized for the case of 
nonuniform samples, that is, samples not taken equally spaced in time. 
The Shannon sampling theory for non-uniform sampling states that a 
band-limited signal can be perfectly reconstructed from its samples if 
the average sampling rate satisfies the Nyquist condition.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>4<span>]</span></a></sup>
 Therefore, although uniformly spaced samples may result in easier 
reconstruction algorithms, it is not a necessary condition for perfect 
reconstruction.</p>
<p>The general theory for non-baseband and nonuniform samples was developed in 1967 by <a href="http://en.wikipedia.org/wiki/Henry_Landau" title="Henry Landau">Landau</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>5<span>]</span></a></sup> He proved that, to paraphrase roughly, the average sampling rate (uniform or otherwise) must be twice the <i>occupied</i> bandwidth of the signal, assuming it is <i>a priori</i>
 known what portion of the spectrum was occupied. In the late 1990s, 
this work was partially extended to cover signals of when the amount of 
occupied bandwidth was known, but the actual occupied portion of the 
spectrum was unknown.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>6<span>]</span></a></sup> In the 2000s, a complete theory was developed (see the section <a href="http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem#Beyond_Nyquist" title="Nyquist–Shannon sampling theorem">Beyond Nyquist</a> below) using <a href="http://en.wikipedia.org/wiki/Compressed_sensing" title="Compressed sensing">compressed sensing</a>. In particular, the theory, using signal processing language, is described in this 2009 paper.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>7<span>]</span></a></sup>
 They show, among other things, that if the frequency locations are 
unknown, then it is necessary to sample at least at twice the Nyquist 
criteria; in other words, you must pay at least a factor of 2 for not 
knowing the location of the <a href="http://en.wikipedia.org/wiki/Spectrum" title="Spectrum">spectrum</a>. Note that minimum sampling requirements do not necessarily guarantee <a href="http://en.wikipedia.org/wiki/Numerical_stability" title="Numerical stability">stability</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=13" title="Edit section: Beyond Nyquist">edit</a>]</span> <span class="mw-headline" id="Beyond_Nyquist">Beyond Nyquist</span></h2>
<p>The Nyquist–Shannon sampling theorem provides a <a href="http://en.wikipedia.org/wiki/Necessary_and_sufficient_condition" title="Necessary and sufficient condition">sufficient condition</a> for the sampling and reconstruction of a band-limited signal. When reconstruction is done via the <a href="http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">Whittaker–Shannon interpolation formula</a>,
 the Nyquist criterion is also a necessary condition to avoid aliasing, 
in the sense that if samples are taken at a slower rate than twice the 
band limit, then there are some signals that will not be correctly 
reconstructed. However, if further restrictions are imposed on the 
signal, then the Nyquist criterion may no longer be a <a href="http://en.wikipedia.org/wiki/Necessary_and_sufficient_condition" title="Necessary and sufficient condition">necessary condition</a>.</p>
<p>A non-trivial example of exploiting extra assumptions about the signal is given by the recent field of <a href="http://en.wikipedia.org/wiki/Compressed_sensing" title="Compressed sensing">compressed sensing</a>,
 which allows for full reconstruction with a sub-Nyquist sampling rate. 
Specifically, this applies to signals that are sparse (or compressible) 
in some domain. As an example, compressed sensing deals with signals 
that may have a low over-all bandwidth (say, the <i>effective</i> bandwidth <span class="texhtml"><i>E</i><i>B</i></span>), but the frequency locations are unknown, rather than all together in a single band, so that the <a href="http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem#Sampling_of_non-baseband_signals" title="Nyquist–Shannon sampling theorem">passband technique</a> doesn't apply. In other words, the frequency spectrum is sparse. Traditionally, the necessary sampling rate is thus <span class="texhtml"><i>B</i> / 2</span>.
 Using compressed sensing techniques, the signal could be perfectly 
reconstructed if it is sampled at a rate slightly greater than the <span class="texhtml"><i>E</i><i>B</i> / 2</span>. The downside of this approach is that reconstruction is no longer given by a formula, but instead by the solution to a <a href="http://en.wikipedia.org/wiki/Convex_optimization" title="Convex optimization">convex optimization program</a> which requires well-studied but nonlinear methods.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=14" title="Edit section: Historical background">edit</a>]</span> <span class="mw-headline" id="Historical_background">Historical background</span></h2>
<p>The <b>sampling theorem</b> was implied by the work of <a href="http://en.wikipedia.org/wiki/Harry_Nyquist" title="Harry Nyquist">Harry Nyquist</a> in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2<i>B</i> independent pulse samples could be sent through a system of bandwidth <i>B</i>; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, <a href="http://en.wikipedia.org/wiki/Karl_K%C3%BCpfm%C3%BCller" title="Karl Küpfmüller">Karl Küpfmüller</a> showed a similar result,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>8<span>]</span></a></sup> and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response <i><a href="http://en.wikipedia.org/wiki/Sine_integral" title="Sine integral" class="mw-redirect">Integralsinus</a></i>; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a <i>Küpfmüller filter</i> (but seldom so in English).</p>
<p>The sampling theorem, essentially a <a href="http://en.wikipedia.org/wiki/Dual" title="Dual">dual</a> of Nyquist's result, was proved by <a href="http://en.wikipedia.org/wiki/Claude_E._Shannon" title="Claude E. Shannon" class="mw-redirect">Claude E. Shannon</a> in 1949 ("Communication in the presence of noise"). <a href="http://en.wikipedia.org/wiki/V._A._Kotelnikov" title="V. A. Kotelnikov" class="mw-redirect">V. A. Kotelnikov</a>
 published similar results in 1933 ("On the transmission capacity of the
 'ether' and of cables in electrical communications", translation from 
the Russian), as did the mathematician <a href="http://en.wikipedia.org/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a>
 in 1915 ("Expansions of the Interpolation-Theory", "Theorie der 
Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function 
theory"), and <a href="http://en.wikipedia.org/wiki/Dennis_Gabor" title="Dennis Gabor">Gabor</a> in 1946 ("Theory of communication").</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=15" title="Edit section: Other discoverers">edit</a>]</span> <span class="mw-headline" id="Other_discoverers">Other discoverers</span></h3>
<p>Others who have independently discovered or played roles in the 
development of the sampling theorem have been discussed in several 
historical articles, for example by Jerri<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span>[</span>9<span>]</span></a></sup> and by Lüke.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span>[</span>10<span>]</span></a></sup>
 For example, Lüke points out that H. Raabe, an assistant to Küpfmüller,
 proved the theorem in his 1939 Ph.D. dissertation; the term <i>Raabe condition</i> came to be associated with the criterion for unambiguous representation (sampling rate greater than twice the bandwidth).</p>
<p>Meijering<sup id="cite_ref-EM_10-0" class="reference"><a href="#cite_note-EM-10"><span>[</span>11<span>]</span></a></sup> mentions several other discoverers and names in a paragraph and pair of footnotes:</p>
<blockquote>
<p>As pointed out by Higgins [135], the sampling theorem should really 
be considered in two parts, as done above: the first stating the fact 
that a bandlimited function is completely determined by its samples, the
 second describing how to reconstruct the function using its samples. 
Both parts of the sampling theorem were given in a somewhat different 
form by J. M. Whittaker [350, 351, 353] and before him also by Ogura 
[241, 242]. They were probably not aware of the fact that the first part
 of the theorem had been stated as early as 1897 by Borel [25].<sup>27</sup>
 As we have seen, Borel also used around that time what became known as 
the cardinal series. However, he appears not to have made the link 
[135]. In later years it became known that the sampling theorem had been
 presented before Shannon to the Russian communication community by 
Kotel'nikov [173]. In more implicit, verbal form, it had also been 
described in the German literature by Raabe [257]. Several authors [33, 
205] have mentioned that Someya [296] introduced the theorem in the 
Japanese literature parallel to Shannon. In the English literature, 
Weston [347] introduced it independently of Shannon around the same 
time.<sup>28</sup></p>
</blockquote>
<blockquote>
<p><sup>27</sup> Several authors, following Black [16], have claimed 
that this first part of the sampling theorem was stated even earlier by 
Cauchy, in a paper [41] published in 1841. However, the paper of Cauchy 
does not contain such a statement, as has been pointed out by Higgins 
[135].</p>
</blockquote>
<blockquote>
<p><sup>28</sup> As a consequence of the discovery of the several 
independent introductions of the sampling theorem, people started to 
refer to the theorem by including the names of the aforementioned 
authors, resulting in such catchphrases as “the 
Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem" [155] or even "the
 Whittaker-Kotel'nikov-Raabe-Shannon-Someya sampling theorem" [33]. To 
avoid confusion, perhaps the best thing to do is to refer to it as the 
sampling theorem, "rather than trying to find a title that does justice 
to all claimants" [136].</p>
</blockquote>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=16" title="Edit section: Why Nyquist?">edit</a>]</span> <span class="mw-headline" id="Why_Nyquist.3F">Why Nyquist?</span></h3>
<p>Exactly how, when, or why <a href="http://en.wikipedia.org/wiki/Harry_Nyquist" title="Harry Nyquist">Harry Nyquist</a> had his name attached to the sampling theorem remains obscure. The term <i>Nyquist Sampling Theorem</i> (capitalized thus) appeared as early as 1959 in a book from his former employer, <a href="http://en.wikipedia.org/wiki/Bell_Labs" title="Bell Labs">Bell Labs</a>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span>[</span>12<span>]</span></a></sup> and appeared again in 1963,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span>[</span>13<span>]</span></a></sup> and not capitalized in 1965.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span>[</span>14<span>]</span></a></sup> It had been called the <i>Shannon Sampling Theorem</i> as early as 1954,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span>[</span>15<span>]</span></a></sup> but also just <i>the sampling theorem</i> by several other books in the early 1950s.</p>
<p>In 1958, Blackman and Tukey<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span>[</span>16<span>]</span></a></sup> cited Nyquist's 1928 paper as a reference for <i>the sampling theorem of information theory</i>,
 even though that paper does not treat sampling and reconstruction of 
continuous signals as others did. Their glossary of terms includes these
 entries:</p>
<dl>
<dd><i>Sampling theorem</i> (<i>of information theory</i>)</dd>
</dl>
<dl>
<dd>Nyquist's result that equi-spaced data, with two or more points per 
cycle of highest frequency, allows reconstruction of band-limited 
functions. (See <i>Cardinal theorem.</i>)</dd>
</dl>
<dl>
<dd><i>Cardinal theorem</i> (<i>of interpolation theory</i>)</dd>
</dl>
<dl>
<dd>A precise statement of the conditions under which values given at a 
doubly infinite set of equally spaced points can be interpolated to 
yield a continuous band-limited function with the aid of the function</dd>
</dl>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="\frac{\sin (x - x_i)}{x - x_i}." src="wikipedia-Nyquist%E2%80%93Shannon_sampling_theorem_pliki/af8c24a20d30cc4f3c4814e6f10532cb.png"></dd>
</dl>
</dd>
</dl>
<p>Exactly what "Nyquist's result" they are referring to remains mysterious.</p>
<p>When Shannon stated and proved the sampling theorem in his 1949 paper, according to Meijering<sup id="cite_ref-EM_10-1" class="reference"><a href="#cite_note-EM-10"><span>[</span>11<span>]</span></a></sup> "he referred to the critical sampling interval <i>T</i> = 1/(2<i>W</i>) as the <i>Nyquist interval</i> corresponding to the band <i>W</i>,
 in recognition of Nyquist’s discovery of the fundamental importance of 
this interval in connection with telegraphy." This explains Nyquist's 
name on the critical interval, but not on the theorem.</p>
<p>Similarly, Nyquist's name was attached to <i><a href="http://en.wikipedia.org/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a></i> in 1953 by <a href="http://en.wikipedia.org/wiki/Harold_Stephen_Black" title="Harold Stephen Black">Harold S. Black</a>:<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span>[</span>17<span>]</span></a></sup></p>
<dl>
<dd>"If the essential frequency range is limited to <i>B</i> cycles per second, 2<i>B</i>
 was given by Nyquist as the maximum number of code elements per second 
that could be unambiguously resolved, assuming the peak interference is 
less half a quantum step. This rate is generally referred to as <b>signaling at the Nyquist rate</b> and 1/(2<i>B</i>) has been termed a <i>Nyquist interval</i>." (bold added for emphasis; italics as in the original)</dd>
</dl>
<p>According to the <a href="http://en.wikipedia.org/wiki/OED" title="OED" class="mw-redirect">OED</a>, this may be the origin of the term <i>Nyquist rate</i>. In Black's usage, it is not a sampling rate, but a signaling rate.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=17" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Hartley%27s_law" title="Hartley's law" class="mw-redirect">Hartley's law</a></li>
<li><a href="http://en.wikipedia.org/wiki/Reconstruction_from_zero_crossings" title="Reconstruction from zero crossings">Reconstruction from zero crossings</a></li>
<li><a href="http://en.wikipedia.org/wiki/Zero-order_hold" title="Zero-order hold">Zero-order hold</a></li>
<li>The <a href="http://en.wikipedia.org/wiki/Cheung%E2%80%93Marks_theorem" title="Cheung–Marks theorem">Cheung–Marks theorem</a> specifies conditions where restoration of a signal by the sampling theorem can become ill-posed.</li>
<li><a href="http://en.wikipedia.org/wiki/Balian%E2%80%93Low_theorem" title="Balian–Low theorem">Balian–Low theorem</a>, a similar theoretical lower-bound on sampling rates, but which applies to <a href="http://en.wikipedia.org/wiki/Time%E2%80%93frequency_analysis" title="Time–frequency analysis">time–frequency transforms</a>.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=18" title="Edit section: Notes">edit</a>]</span> <span class="mw-headline" id="Notes">Notes</span></h2>
<div class="reflist references-column-width" style="column-width: 30em; -moz-column-width: 30em; -webkit-column-width: 30em; list-style-type: decimal;">
<ol class="references">
<li id="cite_note-Shannon49-0">^ <a href="#cite_ref-Shannon49_0-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Shannon49_0-1"><sup><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Claude_E._Shannon" title="Claude E. Shannon" class="mw-redirect">C. E. Shannon</a>, "Communication in the presence of noise", <a href="http://en.wikipedia.org/wiki/Proceedings_of_the_IRE" title="Proceedings of the IRE" class="mw-redirect">Proc.&nbsp;Institute of Radio Engineers</a>, vol.&nbsp;37, no.&nbsp;1, pp.&nbsp;10–21, Jan.&nbsp;1949. <a href="http://www.stanford.edu/class/ee104/shannonpaper.pdf" class="external text" rel="nofollow">Reprint as classic paper in: <i>Proc.&nbsp;IEEE</i>, vol.&nbsp;86, no.&nbsp;2, (Feb.&nbsp;1998)</a></li>
<li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> supp(<i>X</i>) means the <a href="http://en.wikipedia.org/wiki/Support_%28mathematics%29" title="Support (mathematics)">support</a> of X.</li>
<li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> The time-domain form follows from rows 202 and 102 of the <a href="http://en.wikipedia.org/wiki/Fourier_transform#Tables_of_important_Fourier_transforms" title="Fourier transform">transform tables</a></li>
<li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> Nonuniform Sampling, Theory and Practice (ed. F. Marvasti), Kluwer Academic/Plenum Publishers, New York, 2000</li>
<li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> H. J. Landau, 
“Necessary density conditions for sampling and interpolation of certain 
entire functions,” Acta Math., vol. 117, pp.37–52, Feb. 1967.</li>
<li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> see, e.g., P. 
Feng, “Universal minimum-rate sampling and spectrum-blind reconstruction
 for multiband signals,” Ph.D. dissertation, University of Illinois at 
Urbana-Champaign, 1997.</li>
<li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.154.4255" class="external text" rel="nofollow">Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals</a>, Moshe Mishali and Yonina C. Eldar, in <b>IEEE Trans. Signal Processing</b>, March 2009, Vol 57 Issue 3</li>
<li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <a href="http://en.wikipedia.org/wiki/Karl_K%C3%BCpfm%C3%BCller" title="Karl Küpfmüller">K. Küpfmüller</a>, "Über die Dynamik der selbsttätigen Verstärkungsregler", <i>Elektrische Nachrichtentechnik</i>, vol. 5, no. 11, pp. 459-467, 1928. (German)<br>
&nbsp;&nbsp;K. Küpfmüller, <a href="http://ict.open.ac.uk/classics/2.pdf" class="external text" rel="nofollow">On the dynamics of automatic gain controllers</a>, <i>Elektrische Nachrichtentechnik</i>, vol. 5, no. 11, pp. 459-467, 1928 (English translation 2005).</li>
<li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <a href="http://en.wikipedia.org/wiki/Abdul_Jerri" title="Abdul Jerri">Abdul Jerri</a>, <a href="http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1455040" class="external text" rel="nofollow">The Shannon Sampling Theorem—Its Various Extensions and Applications: A Tutorial Review</a>, <i>Proceedings of the IEEE</i>, 65:1565–1595, Nov. 1977. See also <a href="http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1455576" class="external text" rel="nofollow">Correction to "The Shannon sampling theorem—Its various extensions and applications: A tutorial review"</a>, Proceedings of the IEEE, 67:695, April 1979</li>
<li id="cite_note-9"><b><a href="#cite_ref-9">^</a></b> Hans Dieter Lüke, <a href="http://dx.doi.org/10.1109/35.755459" class="external text" rel="nofollow">The Origins of the Sampling Theorem</a>, <i>IEEE Communications Magazine,</i> pp.106–108, April 1999.</li>
<li id="cite_note-EM-10">^ <a href="#cite_ref-EM_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-EM_10-1"><sup><i><b>b</b></i></sup></a> Erik Meijering, <a href="http://dx.doi.org/10.1109/5.993400" class="external text" rel="nofollow">A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing</a>, <i><a href="http://en.wikipedia.org/wiki/Proc._IEEE" title="Proc. IEEE" class="mw-redirect">Proc. IEEE</a>,</i> 90, 2002.</li>
<li id="cite_note-11"><b><a href="#cite_ref-11">^</a></b> <span class="citation book">Members of the Technical Staff of Bell Telephone Lababoratories (1959). <i>Transmission Systems for Communications</i>. AT&amp;T. pp.&nbsp;26–4 (Vol.2).</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Transmission+Systems+for+Communications&amp;rft.aulast=Members+of+the+Technical+Staff+of+Bell+Telephone+Lababoratories&amp;rft.au=Members+of+the+Technical+Staff+of+Bell+Telephone+Lababoratories&amp;rft.date=1959&amp;rft.pages=pp.%26nbsp%3B26%E2%80%934+%28Vol.2%29&amp;rft.pub=AT%26T&amp;rfr_id=info:sid/en.wikipedia.org:Nyquist%E2%80%93Shannon_sampling_theorem"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-12"><b><a href="#cite_ref-12">^</a></b> <span class="citation book">Ernst Adolph Guillemin (1963). <a href="http://books.google.com/books?id=jtI-AAAAIAAJ&amp;q=Nyquist-sampling-theorem+date:0-1965&amp;dq=Nyquist-sampling-theorem+date:0-1965&amp;as_brr=0&amp;ei=ief5Ru-DJqegowLyiJTpBw&amp;pgis=1" class="external text" rel="nofollow"><i>Theory of Linear Physical Systems</i></a>. Wiley<span class="printonly">. <a href="http://books.google.com/books?id=jtI-AAAAIAAJ&amp;q=Nyquist-sampling-theorem+date:0-1965&amp;dq=Nyquist-sampling-theorem+date:0-1965&amp;as_brr=0&amp;ei=ief5Ru-DJqegowLyiJTpBw&amp;pgis=1" class="external free" rel="nofollow">http://books.google.com/books?id=jtI-AAAAIAAJ&amp;q=Nyquist-sampling-theorem+date:0-1965&amp;dq=Nyquist-sampling-theorem+date:0-1965&amp;as_brr=0&amp;ei=ief5Ru-DJqegowLyiJTpBw&amp;pgis=1</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Theory+of+Linear+Physical+Systems&amp;rft.aulast=Ernst+Adolph+Guillemin&amp;rft.au=Ernst+Adolph+Guillemin&amp;rft.date=1963&amp;rft.pub=Wiley&amp;rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjtI-AAAAIAAJ%26q%3DNyquist-sampling-theorem%2Bdate%3A0-1965%26dq%3DNyquist-sampling-theorem%2Bdate%3A0-1965%26as_brr%3D0%26ei%3Dief5Ru-DJqegowLyiJTpBw%26pgis%3D1&amp;rfr_id=info:sid/en.wikipedia.org:Nyquist%E2%80%93Shannon_sampling_theorem"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-13"><b><a href="#cite_ref-13">^</a></b> Richard A. Roberts and Ben F. Barton, <i>Theory of Signal Detectability: Composite Deferred Decision Theory</i>, 1965.</li>
<li id="cite_note-14"><b><a href="#cite_ref-14">^</a></b> Truman S. Gray, <i>Applied Electronics: A First Course in Electronics, Electron Tubes, and Associated Circuits</i>, 1954.</li>
<li id="cite_note-15"><b><a href="#cite_ref-15">^</a></b> R. B. Blackman and J. W. Tukey, <i>The Measurement of Power Spectra&nbsp;: From the Point of View of Communications Engineering</i>, New York: Dover, 1958.</li>
<li id="cite_note-16"><b><a href="#cite_ref-16">^</a></b> Harold S. Black, <i>Modulation Theory,</i> 1953</li>
</ol>
</div>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=19" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a>,
 "On the Functions Which are Represented by the Expansions of the 
Interpolation Theory", Proc. Royal Soc. Edinburgh, Sec. A, vol.35, 
pp.&nbsp;181–194, 1915</li>
<li><a href="http://en.wikipedia.org/wiki/Harry_Nyquist" title="Harry Nyquist">H. Nyquist</a>, "Certain topics in telegraph transmission theory", Trans. AIEE, vol. 47, pp.&nbsp;617–644, Apr. 1928 <a href="http://replay.web.archive.org/20060706192816/http://www.loe.ee.upatras.gr/Comes/Notes/Nyquist.pdf" class="external text" rel="nofollow">Reprint as classic paper in: <i>Proc. IEEE, Vol. 90, No. 2, Feb 2002</i></a>.</li>
<li><a href="http://en.wikipedia.org/wiki/Karl_K%C3%BCpfm%C3%BCller" title="Karl Küpfmüller">Karl Küpfmüller</a>,
 "Utjämningsförlopp inom Telegraf- och Telefontekniken", ("Transients in
 telegraph and telephone engineering"), Teknisk Tidskrift, no. 9 
pp.&nbsp;153–160 and 10 pp.&nbsp;178–182, 1931. <a href="http://runeberg.org/tektid/1931e/0157.html" class="external autonumber" rel="nofollow">[1]</a> <a href="http://runeberg.org/tektid/1931e/0182.html" class="external autonumber" rel="nofollow">[2]</a></li>
<li><a href="http://en.wikipedia.org/wiki/V._A._Kotelnikov" title="V. A. Kotelnikov" class="mw-redirect">V. A. Kotelnikov</a>,
 "On the carrying capacity of the ether and wire in telecommunications",
 Material for the First All-Union Conference on Questions of 
Communication, Izd. Red. Upr. Svyazi RKKA, Moscow, 1933 (Russian). <a href="http://ict.open.ac.uk/classics/1.pdf" class="external text" rel="nofollow">(english translation, PDF)</a></li>
<li><a href="http://en.wikipedia.org/wiki/J._M._Whittaker" title="J. M. Whittaker" class="mw-redirect">J. M. Whittaker</a>, <i>Interpolatory Function Theory</i>, Cambridge Univ. Press, Cambridge, England, 1935.</li>
<li><a href="http://en.wikipedia.org/wiki/Claude_E._Shannon" title="Claude E. Shannon" class="mw-redirect">C. E. Shannon</a>, "Communication in the presence of noise", Proc. Institute of Radio Engineers, vol. 37, no.1, pp.&nbsp;10–21, Jan. 1949. <a href="http://www.stanford.edu/class/ee104/shannonpaper.pdf" class="external text" rel="nofollow">Reprint as classic paper in: <i>Proc. IEEE</i>, Vol. 86, No. 2, (Feb 1998)</a></li>
<li>J. R. Higgins: <i>Five short stories about the cardinal series</i>, Bulletin of the AMS 12(1985)</li>
<li>R.J. Marks II: <i><a href="http://marksmannet.com/RobertMarks/REPRINTS/1999_IntroductionToShannonSamplingAndInterpolationTheory.pdf" class="external text" rel="nofollow">Introduction to Shannon Sampling and Interpolation Theory</a></i>, Spinger-Verlag, 1991.</li>
<li>R.J. Marks II, Editor: <a href="http://marksmannet.com/RobertMarks/REPRINTS/1993_AdvancedTopicsOnShannon.pdf" class="external text" rel="nofollow">Advanced Topics in Shannon Sampling and Interpolation Theory</a>, Springer-Verlag, 1993.</li>
<li>R.J. Marks II, <i>Handbook of Fourier Analysis and Its Applications,</i> Oxford University Press, (2009), Chapters 5-8. <a href="http://books.google.com/books?id=Sp7O4bocjPAC" class="external text" rel="nofollow">Google books</a>.</li>
<li>Michael Unser: <i><a href="http://bigwww.epfl.ch/publications/unser0001.html" class="external text" rel="nofollow">Sampling-50 Years after Shannon</a></i>, Proc. IEEE, vol. 88, no. 4, pp.&nbsp;569–587, April 2000</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&amp;action=edit&amp;section=20" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a href="http://www.vias.org/simulations/simusoft_nykvist.html" class="external text" rel="nofollow">Learning by Simulations</a> Interactive simulation of the effects of inadequate sampling</li>
<li><a href="http://spazioscuola.altervista.org/UndersamplingAR/UndersamplingARnv.htm" class="external text" rel="nofollow">Undersampling and an application of it</a></li>
<li><a href="http://www.lavryengineering.com/documents/Sampling_Theory.pdf" class="external text" rel="nofollow">Sampling Theory For Digital Audio</a></li>
<li><a href="http://www.stsip.org/" class="external text" rel="nofollow">Journal devoted to Sampling Theory</a></li>
<li><a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.163.2887&amp;rep=rep1&amp;type=pdf" class="external text" rel="nofollow">"The Origins of the Sampling Theorem" by Hans Dieter Lüke published in "IEEE Communications Magazine" April 1999</a></li>
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<td class="navbox-group" style="">Techniques</td>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Impulse_invariance" title="Impulse invariance">Impulse invariance</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Bilinear_transform" title="Bilinear transform">bilinear transform</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Pole%E2%80%93zero_mapping" title="Pole–zero mapping" class="mw-redirect">pole–zero mapping</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Z-transform" title="Z-transform">Z-transform</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Advanced_Z-transform" title="Advanced Z-transform">advanced Z-transform</a></span></div>
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<td class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling</a></td>
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<div style="padding:0em 0.25em"><span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Oversampling" title="Oversampling">oversampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Undersampling" title="Undersampling">undersampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Downsampling" title="Downsampling">downsampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Upsampling" title="Upsampling">upsampling</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">aliasing</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Sampling_rate" title="Sampling rate">sampling rate</a>&nbsp;<b>·</b></span> <span style="white-space:nowrap;"><a href="http://en.wikipedia.org/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a>/<a href="http://en.wikipedia.org/wiki/Nyquist_frequency" title="Nyquist frequency">frequency</a></span></div>
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<th style="" colspan="2" class="navbox-title"><span class="collapseButton">[<a href="#" id="collapseButton1">show</a>]</span><span style="float:left;width:6em;text-align:left;"><span class="noprint plainlinks navbar" style=""><span style="white-space:nowrap;word-spacing:-.12em;"><a href="http://en.wikipedia.org/wiki/Template:Compression_methods" title="Template:Compression methods"><span style=";;background:none transparent;border:none;font-size:100%;" title="View this template">v</span></a> <span style=";;background:none transparent;border:none;font-size:100%;"><b>·</b></span> <a href="http://en.wikipedia.org/wiki/Template_talk:Compression_methods" title="Template talk:Compression methods"><span style=";;background:none transparent;border:none;font-size:100%;" title="Discuss this template">d</span></a> <span style=";;background:none transparent;border:none;font-size:100%;"><b>·</b></span> <a href="http://en.wikipedia.org/w/index.php?title=Template:Compression_methods&amp;action=edit" class="external text" rel="nofollow"><span style=";;background:none transparent;border:none;font-size:100%;" title="Edit this template">e</span></a></span></span></span><span class="" style="font-size:110%;"><a href="http://en.wikipedia.org/wiki/Data_compression" title="Data compression">Data compression</a> methods</span></th>
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<td class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Lossless_data_compression" title="Lossless data compression">Lossless</a></td>
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<div style="padding:0em 0.75em;"><a href="http://en.wikipedia.org/wiki/Information_theory" title="Information theory">Theory</a></div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29" title="Entropy (information theory)">Entropy</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Complexity</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Redundancy_%28information_theory%29" title="Redundancy (information theory)">Redundancy</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Lossy_compression" title="Lossy compression">Lossy</a></div>
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<div style="padding:0em 0.75em;"><a href="http://en.wikipedia.org/wiki/Entropy_encoding" title="Entropy encoding">Entropy encoding</a></div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/w/index.php?title=Shannon%E2%80%93Fano%E2%80%93Elias_coding&amp;action=edit&amp;redlink=1" class="new" title="Shannon–Fano–Elias coding (page does not exist)">Shannon–Fano–Elias</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Huffman_coding" title="Huffman coding">Huffman</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive Huffman</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Arithmetic_coding" title="Arithmetic coding">Arithmetic</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Range_encoding" title="Range encoding">Range</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Golomb_coding" title="Golomb coding">Golomb</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Universal_code_%28data_compression%29" title="Universal code (data compression)">Universal</a> (<a href="http://en.wikipedia.org/wiki/Elias_gamma_coding" title="Elias gamma coding">Gamma</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Fibonacci_coding" title="Fibonacci coding">Fibonacci</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Levenshtein_coding" title="Levenshtein coding" class="mw-redirect">Levenshtein</a>)</div>
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<div style="padding:0em 0.75em;"><a href="http://en.wikipedia.org/wiki/Dictionary_coder" title="Dictionary coder">Dictionary</a></div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Run-length_encoding" title="Run-length encoding">RLE</a>&nbsp;<span style="font-weight:bold;">·</span> <a href="http://en.wikipedia.org/wiki/Byte_pair_encoding" title="Byte pair encoding">Byte pair encoding</a>&nbsp;<span style="font-weight:bold;">·</span> <a href="http://en.wikipedia.org/wiki/DEFLATE" title="DEFLATE">DEFLATE</a>&nbsp;<span style="font-weight:bold;">·</span> <a href="http://en.wikipedia.org/wiki/Lempel%E2%80%93Ziv" title="Lempel–Ziv" class="mw-redirect">Lempel–Ziv</a> (<a href="http://en.wikipedia.org/wiki/LZ77_and_LZ78" title="LZ77 and LZ78">LZ77/78</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Lempel%E2%80%93Ziv%E2%80%93Storer%E2%80%93Szymanski" title="Lempel–Ziv–Storer–Szymanski">LZSS</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Lempel%E2%80%93Ziv%E2%80%93Welch" title="Lempel–Ziv–Welch">LZW</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/LZWL" title="LZWL">LZWL</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Lempel%E2%80%93Ziv%E2%80%93Oberhumer" title="Lempel–Ziv–Oberhumer">LZO</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Lempel%E2%80%93Ziv%E2%80%93Markov_chain_algorithm" title="Lempel–Ziv–Markov chain algorithm">LZMA</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/LZX_%28algorithm%29" title="LZX (algorithm)">LZX</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/LZRW" title="LZRW">LZRW</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/LZJB" title="LZJB">LZJB</a>&nbsp;<span style="font-weight:bold;">·</span> <a href="http://en.wikipedia.org/wiki/Lempel%E2%80%93Ziv%E2%80%93Stac" title="Lempel–Ziv–Stac">LZS</a>&nbsp;<span style="font-weight:bold;">·</span> <a href="http://en.wikipedia.org/w/index.php?title=Lempel%E2%80%93Ziv%E2%80%93Tamayo&amp;action=edit&amp;redlink=1" class="new" title="Lempel–Ziv–Tamayo (page does not exist)">LZT</a>&nbsp;<span style="font-weight:bold;">·</span> <a href="http://en.wikipedia.org/w/index.php?title=Reduced_Offset_Lempel_Ziv&amp;action=edit&amp;redlink=1" class="new" title="Reduced Offset Lempel Ziv (page does not exist)">ROLZ</a>)&nbsp;<span style="font-weight:bold;">·</span> <a href="http://en.wikipedia.org/wiki/Statistical_Lempel_Ziv" title="Statistical Lempel Ziv">Statistical Lempel Ziv</a></div>
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<div style="padding:0em 0.75em;">Others</div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Context_tree_weighting" title="Context tree weighting">CTW</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Burrows%E2%80%93Wheeler_transform" title="Burrows–Wheeler transform">BWT</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Prediction_by_Partial_Matching" title="Prediction by Partial Matching" class="mw-redirect">PPM</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Dynamic_Markov_compression" title="Dynamic Markov compression">DMC</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Delta_encoding" title="Delta encoding">Delta</a></div>
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<td class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Audio_compression_%28data%29" title="Audio compression (data)">Audio</a></td>
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<div style="padding:0em 0.75em;"><a href="http://en.wikipedia.org/wiki/Acoustics" title="Acoustics">Theory</a></div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Companding" title="Companding">Companding</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">Convolution</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Dynamic_range" title="Dynamic range">Dynamic range</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Latency_%28audio%29" title="Latency (audio)">Latency</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling</a>&nbsp;<span style="font-weight:bold;">·</span>  <strong class="selflink">Nyquist–Shannon theorem</strong>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Sound_quality" title="Sound quality">Sound quality</a></div>
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<div style="padding:0em 0.75em;"><a href="http://en.wikipedia.org/wiki/Audio_codec" title="Audio codec">Audio codec</a> parts</div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> (<a href="http://en.wikipedia.org/wiki/Log_area_ratio" title="Log area ratio">LAR</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a>)&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Algebraic_Code_Excited_Linear_Prediction" title="Algebraic Code Excited Linear Prediction">ACELP</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/A-law_algorithm" title="A-law algorithm">A-law</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/%CE%9C-law_algorithm" title="Μ-law algorithm">μ-law</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Adaptive_DPCM" title="Adaptive DPCM">ADPCM</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/DPCM" title="DPCM">DPCM</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></div>
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<div style="padding:0em 0.75em;">Others</div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Bit_rate" title="Bit rate">Bit rate</a> (<a href="http://en.wikipedia.org/wiki/Constant_bitrate" title="Constant bitrate">CBR</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Average_bitrate" title="Average bitrate">ABR</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Variable_bitrate" title="Variable bitrate">VBR</a>)&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Speech_encoding" title="Speech encoding" class="mw-redirect">Speech compression</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Sub-band_coding" title="Sub-band coding">Sub-band coding</a></div>
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<td class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Image_compression" title="Image compression">Image</a></td>
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<div style="padding:0em 0.75em;">Terms</div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Color_space" title="Color space">Color space</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Pixel" title="Pixel">Pixel</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Compression_artifact" title="Compression artifact">Compression artifact</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Image_resolution" title="Image resolution">Image resolution</a></div>
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<div style="padding:0em 0.75em;">Methods</div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Run-length_encoding" title="Run-length encoding">RLE</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Fractal_compression" title="Fractal compression">Fractal</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Wavelet_compression" title="Wavelet compression" class="mw-redirect">Wavelet</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/EZW" title="EZW" class="mw-redirect">EZW</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Pyramid_%28image_processing%29" title="Pyramid (image processing)">LP</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Chain_code" title="Chain code">Chain code</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Karhunen-Lo%C3%A8ve_transform" title="Karhunen-Loève transform" class="mw-redirect">KLT</a></div>
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<div style="padding:0em 0.75em;">Others</div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Standard_test_image" title="Standard test image">Test images</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio" title="Peak signal-to-noise ratio">PSNR quality measure</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Quantization_%28image_processing%29" title="Quantization (image processing)">Quantization</a></div>
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<td class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Video_compression" title="Video compression">Video</a></td>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Video#Characteristics_of_video_streams" title="Video">Video characteristics</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Film_frame" title="Film frame">Frame</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Frame_rate" title="Frame rate">Frame rate</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Interlaced_video" title="Interlaced video">Interlace</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Video_compression_picture_types" title="Video compression picture types">Frame types</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Video_quality" title="Video quality">Video quality</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Video_resolution" title="Video resolution" class="mw-redirect">Video resolution</a></div>
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<div style="padding:0em 0.75em;"><a href="http://en.wikipedia.org/wiki/Video_codec" title="Video codec">Video codec parts</a></div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Motion_compensation" title="Motion compensation">Motion compensation</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Quantization_%28signal_processing%29" title="Quantization (signal processing)">Quantization</a></div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Video_codec" title="Video codec">Video codecs</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Rate_distortion_theory" title="Rate distortion theory" class="mw-redirect">Rate distortion theory</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Bit_rate" title="Bit rate">Bit rate</a> (<a href="http://en.wikipedia.org/wiki/Constant_bitrate" title="Constant bitrate">CBR</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Average_bitrate" title="Average bitrate">ABR</a>&nbsp;<span style="font-weight:bold;">·</span>  <a href="http://en.wikipedia.org/wiki/Variable_bitrate" title="Variable bitrate">VBR</a>)</div>
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<div style="padding:0em 0.25em"><a href="http://en.wikipedia.org/wiki/Timeline_of_information_theory" title="Timeline of information theory">Timeline of information theory, data compression, and error-correcting codes</a></div>
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<td></td>
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<td class="navbox-abovebelow" style="" colspan="2">See <a href="http://en.wikipedia.org/wiki/Template:Compression_formats" title="Template:Compression formats">Compression formats</a> for formats and <a href="http://en.wikipedia.org/wiki/Template:Compression_software_implementations" title="Template:Compression software implementations">Compression software implementations</a> for codecs</td>
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